Engineering Calculations

Transcription

1 Engineering Calculations (EC) Engineering Calculations In order to properly communicate with other engineers / scientists you must use the correct language.how can you use concise language to convey not only what you measured, but how precise the measurement was? Explain the utility of the proper use of units # Define and give examples of value, units, and dimensions in a given expression # Convert from one set of units to another # Identify an invalid equation, based on dimensional arguments # Compare two quantities using a dimensionless group Correctly use scientific notation # Write a value in scientific notation(using the correct number of significant digits) # Determine the number of significant digits both in a given value as well as in the result of arithmetic While measuring data is enormously useful you cannot possibly measure every possible bit of information under every possible condition(it would take too long and would be far too expensive).how can you use limited data to obtain enough information to get by? Utilize curve - fitting or linear interpolation to estimate unmeasured data from measured data "Check" your results by order - of - magnitude estimation, back substitution, etc.

2 EC: Define and give examples of value, units, and dimensions in a given expression You use values, units and dimensions all the time: Grocery List Using units properly 1 carton of milk (value: 1; units: carton; dimensions: volume (length 3 )) 1/4 pound burger (value: 0.25; units: pounds; dimensions:mass) A value is the numerical quantity. For example: 5.2 The units tell what that quantity represents. For example: 5.2 liters. The dimensions are the measurable properties that the units represent. For example: a liter is a unit of volume (units are a specific example of a dimensional quantity). Define and give examples of value, units, and dimensions in a given expression Give some more examples of value, unit, dimension triplets. In scientific apps. -> units play a central role in quantification! Treat units as algebraic entities: add/subtract them, if they are of the same units multiply/divide them anytime Algebraic equation "Unit" equation "Unit" equation

3 EC: Convert from one set of units to another Converting units System of units: SI units Units are quite helpful: What does "three rivers stadium is 4 away" mean? 4 what!? Standardized units are more helpful: "Three rivers stadium is 4 miles away" is more helpful than "three rivers is 15 lengths of my backyard away" SI units are an international standard group of units based on the meter (length), second (time), kilogram (mass), degree Kelvin (temperature), ampere (current), lumen (light intensity), and mole (number of elementary entities). NOTE: The dimensions of each of the SI units is shown in parenthesis. Other conventions: English units, CGS units We will (primarily) use SI, but be familiar with others... Use any that you like (but be consistent!). You can (easily) convert them anyway! (see above) Units with same dimensions -> easily inter-converted What units to use? Unit Conversions Convert from one set of units to another So, if it is 1000 feet between Benedum and the Cathedral

4 If there are feet in a meter, how many meters is one step? Multiple and compound units How old are you? 600,000,000 seconds!? Saying 19 years is a lot easier or you could just say 600 Megaseconds. NOTE: Multiple units are units devised purely for convenience. They can be of two types: new names (1 hour is 3600 seconds), or prefixed units (1000 millimeters is 1 meter). The dimensions of multiple units must be the same as the base unit (hours and seconds and days, etc. are all time units). What units do you use for velocity, volume, pressure? Treat units algebraicly -> moving 1 mile every 2 hours means: Your container is 10 cm wide, 10 cm long, and 10 cm high. The volume is: 1000 cm 3 is often called a liter (for convenience). You may recall from physics: Do these units match? Compound units are units made up through algabraic combination of base units. They are often given special names for convenience (volume: liter; force: Newton (pound); pressure: Pascal). Convert from one set of units to another Can you convert compound units? Let's give it a shot!

5 EC: Identify an invalid equation, based on dimensional arguments Dimensions play an important role -> equation validity! It doesn't make sense to say that: 1 meter = 25 seconds! Dimensional Equations Conversely, we already said that feet = 1 meter is valid. Why? [hint] Dimensional Homogeneity The dimensions on both sides of the equals sign must be the same for an equation to be valid. Another way to say this is that valid equations must be dimensionally homogeneous. It is a good practice to make units the same (via conversion). It is also good practice (and a course requirement!) to show all units throughout a problem to test equation validity! Identify an invalid equation, based on dimensional arguments Which of these equations is dimensionally homogeneous? NOTE: IMPORTANT: It is sometimes simpler to write the units as "vertical fractions" to facilitate canceling. Just because an equation is dimensionally homogeneous does not mean that it is valid! Dimensional considerations act as a first test for validity only!

6 EC: Compare two quantities using a dimensionless group Pure numbers have no dimension: 1, 5.2, (sometimes constants do, however!) Alternately, sometime all units cancel out: 6 ft/1ft = 6 Dimensionless numbers Dimensionless quantities can be either pure numbers, or dimensionless groups (expressions where all the units cancel). What are dimensionless groups good for? For one thing, dimensionless groups are useful for "pure" comparisons. Compare two quantities using a dimensionless group Which is further away, Cleveland (136 miles) or State College (223 km)? Must match units in order to compare (in miles or km?). NOTE: Dimensionless groups can be any expression whose units cancel! Here we consider ratio of lengths and times, but force, pressure, resistance, etc. all work as well! The only thing you can't have is a ratio of things with different dimensions! (Obviously, since they would no longer cancel.)

7 EC: Write a value in scientific notation Very large and very small numbers are cumbersome. one option: 600,000,000s = 600 Ms another: 600,000,000s = 6x10 8 s Scientific Notation Either option (compound units or scientific notation) works for both large and small ( m = 1 micrometer = 1x10-6 m) Write a value in scientific notation (using the correct number of significant digits) Write the number of seconds in a day using scientific notation. Why use scientific notation? Represent the precision at which a value is known. Clock without a second hand -> 10:21 AM Clock with a second hand -> 10:21:08 AM Stop watch -> 10:21:08.12 AM Writing numbers to show sig. figs. Significant Figures with decimal: count from non-zero on left to last digit on right without decimal: count from non-zero on left to last non-zero on right scientific notation: typically only sig. figs. are shown (4.05x10 4 ) -> 3 sig. figs (4.0500x10 4 ) -> 5 sig. figs (1.2x10-3 ) -> 2 sig. figs (1.200x10-3 ) -> 4 sig. figs.

8 EC: Determine the number of significant digits Arithmatic Rules It is not good enough to know (and convey) the precision of measurements. We will almost always have to do something with those numbers, we therefore need to consider how is our precision affected by arithmatic manipulation. multiply and divide -> get lowest number of sig. figs. add or subtract -> the last sig. fig. in each number is compared, keep only the (last) sig. fig. which is furthest left (the "largest") Determine the number of significant digits both in a given value as well as in the result of arithmetic How many significant digits are there in 2.04x10-3? Your Car weighs 2100 lb f You put an 8 lb f bag in the trunk What is the car's weight now? [cheat] You measure a samples mass to be 3kg Calculate the samples weight (multiply by gravity: 9.81 m/s 2 ) What is your answer? [cheat]

9 EC: Utilize curve-fitting or linear interpolation Want process variable -> can't directly measure it? Data Manipulation Fitting Data Calibration involves establishing a relationship between the process variable of interest and the measured quantity Once have C=f(x) -> plot data. How do you use it? Interpolation - get data between existing points. Extrapolation - get data outside existing points. Fit curve. Interpolate:, where x and y are the to-be-determined variables and x_i and y_i are the data on the curve. Need to do more often (or extrapolate) -> develop an expression y=f(x) Straight line -> easy Eqn. of line -> y = ax + b Curve Fitting by inspection ("looks good") least squares analysis -> minimize the sum of (absolute error) 2

10 a - slope -> b - intercept -> Non-linear fitting -> cast non-linear equation [like y=ae bx ] into linear form [like (ln(y) = bx + ln(a)]. Plot one y-dependent quantity [here, ln(y)] versus an x-dependent quantity [here, x]. Utilize curve-fitting or linear interpolation to estimate unmeasured data from measured data How do you "linearize": y=bx 3 [cheat] These two types occur often: IMPORTANT: semi-log (log-linear or linear-log): useful for exponential dependencies log-log: useful for polynomial dependencies Do not perform a "double log", that is, don't plot the ln(y) on log paper.

11 EC: "Check" your results by order-of-magnitude estimation, back substitution, etc. check the numbers check the units How do I know I am right?! back-substitution (recalculate using answer) order-of-magnitude (approximate the math and do it in your head - are you within a factor of 10?) reasonableness - did you get a negative time or mass? a volume bigger than the ocean? is the equation dimensionally valid? does you answer have the correct dimension for what you expected to calculate? "Check" your results by order-of-magnitude estimation, back substitution, etc. If I told you there were 310 million seconds in a year, how would you know if I was telling the truth? (Without using a calculator.)